McKean-Vlasov equations and nonlinear Fokker-Planck equations with critical singular Lorentz kernels

TL;DR

The paper advances the theory of McKean–Vlasov and nonlinear Fokker–Planck equations with critical, singular Lorentz kernels by proving well-posedness in endpoint spaces and establishing a Krylov-class framework that yields conditional uniqueness. It develops new PDE tools, notably a 2D Meyers-type upgrade for backward Kolmogorov equations and Aronson-type kernel estimates, to handle endpoint-critical drifts and obtain a robust probabilistic representation of solutions. In particular, it shows existence and uniqueness in Krylov spaces for MV/NFPE with critical kernels in all $d\ge 2$, and derives nonlinear Markov properties of the associated path laws; it also demonstrates non-uniqueness in higher dimensions under supercritical drifts, highlighting the sharpness of the main results. The results have implications for 2D fluid models like the vorticity formulation of NSE, where the Biot–Savart kernel is critical, and provide a probabilistic lens on nonlinear Markov evolutions in singular regimes.

Abstract

We prove the existence and conditional uniqueness in the Krylov class for SDEs with singular divergence-free drifts in the endpoint critical Lorentz space $L^{\infty}(0,T; L^{d,\infty}(\mathbb{R}^d))$, $d \geqslant 2$, which particularly includes the $2$D Biot-Savart law. The uniqueness result is shown to be optimal in dimensions $d \geqslant 3$, by constructing different martingale solutions in the case of supercritical Lorentz drifts. As a consequence, the well-posedness of McKean-Vlasov equations and nonlinear Fokker-Planck equations with critical singular kernels is derived. In particular, this yields the uniqueness of the $2$D vorticity Navier-Stokes equations even in certain supercritical-scaling spaces. Furthermore, we prove that the path laws of solutions to McKean-Vlasov equations with critical singular kernel form a nonlinear Markov process in the sense of McKean.

Figures (4)

• Figure 1: Proof Strategy when $d = 2$
• Figure 2: Solution paths
• Figure 3: Cube $Q_R(\tau, \xi)$
• Figure 4: Proof Strategy when $d\geqslant 3$

Theorems & Definitions (64)

• Theorem 1.1: Well-posedness of 2D critical SDE
• Theorem 1.2
• Theorem 1.3: Non-uniqueness of supercritical SDEs
• Corollary 1.4: Non-uniqueness of supercritical FPE
• Remark 1.5
• Theorem 1.6: Well-posedness of McKean-Vlasov equations
• Theorem 1.7: Well-posedness of nonlinear Fokker-Planck equations
• Theorem 1.8: Well-posedness of 2D vorticity NSE
• Remark 1.9
• Theorem 1.10: Nonlinear Markov process